RESTRICTIONS ON HOMFLYPT AND KAUFFMAN POLYNOMIALS ARISING FROM LOCAL MOVES SANDY GANZELL, MERCEDES GONZALEZ, CHLOE' MARCUM, NINA RYALLS, AND MARIEL SANTOS Abstract. We study the effects of certain local moves on Homflypt and Kauffman polynomials. We show that all Homflypt (or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants. 1. Introduction The Jones polynomial has been widely studied since its introduction in [5]. Divisibility criteria for the Jones polynomial were first observed by Jones 3 in [6], who proved that 1 − VK is a multiple of (1 − t)(1 − t ) for any knot K. The first author observed in [3] that when two links L1 and L2 differ by specific local moves (e.g., crossing changes, ∆-moves, 4-moves, forbidden moves), we can find a fixed polynomial P such that VL1 − VL2 is always a multiple of P . Additional results of this kind were studied in [11] (Cn-moves) and [1] (double-∆-moves). The present paper follows the ideas of [3] to establish divisibility criteria for the Homflypt and Kauffman polynomials. Specifically, we examine the effect of certain local moves on these invariants to show that the difference of Homflypt polynomials for n-component links is always a multiple of a4 − 2a2 + 1 − a2z2, and the difference of Kauffman polynomials for knots is always a multiple of (a2 + 1)(a2 + az + 1). In Section 2 we define our main terms and recall constructions of the Hom- flypt and Kauffman polynomials that are used throughout the paper. The primary theorem of Section 3 establishes the divisibility criterion described above for Homflypt polynomials. We then study consequences of that re- sult, which reveal properties of the Homflypt polynomial and its derivatives, including a proof that a Homflypt polynomial can never be a nontrivial monomial. In Section 4 we prove the analogous theorems for the Kauffman polynomial. Date: May 26, 2020. 2010 Mathematics Subject Classification. Primary 57M27, 57M25. Key words and phrases. Homflypt, Kauffman polynomial, local move. Supported by NSF grant DMS-1560301. 1 2 GANZELL, GONZALEZ, MARCUM, RYALLS, AND SANTOS 2. Preliminaries A local move on a link diagram is a substitution, inside a prescribed ball, of one subdiagram for another, resulting in another link diagram. Local moves are considered \bidirectional" in the sense that if we permit the substitution of subdiagram X for subdiagram Y , then we also permit the substitution of Y for X. Reidemeister moves are local moves that preserve the link type, whereas a crossing change is a local move that may change the link type. We say that the local move M is an unknotting move if repeated uses of M (together with Reidemeister moves) can transform every knot diagram into a diagram of the unknot. We say that M is an unlinking move if repeated uses of M (together with Reidemeister moves) can transform every n-component link diagram into a diagram of the n-component unlink. For example, the standard crossing change is an unlinking move, but the ∆-move (Figure 1) is Figure 1. The ∆-move. not, since it does not affect the pairwise linking numbers of its components. The ∆-move is, however, an unknotting move, as shown in [10]. Thus we say that every knot is ∆-equivalent to the unknot. We assume the reader is familiar with Kauffman’s bracket construction of the Jones Polynomial [8]. For a diagram L, we denote the bracket of L −1 by hLi 2 Z[A; A ]. Recall that the bracket is a regular isotopy invariant (i.e., invariant under Reidemeister moves II and III). The Jones polynomial 3 −w is then given by VL = VL(A) = (−A ) hLi, where w is the writhe of L. We use A = t−1=4 for the indeterminate, and let d = −A2 − A−2, so that h Li = d hLi. The Homflypt polynomial GL = GL(a; z) is most commonly defined [2] −1 by a skein relation in two variables: aGL+ − a GL− = zGL0 , where GL is an invariant normalized to equal 1 for the unknot. The links L+, L− and L0 as usual are identical except at a single crossing site. However, we may equivalently define GL(a; z) by a bracket hLih according to the rules: hL+ih − hL−ih = z hL0ih h ih = a h ih −1 h ih = a h ih h ih = 1 where hLih is a regular isotopy invariant. The Homflypt polynomial is then −w defined by GL = a hLih, where w is the writhe of L. See [7] for details of a−a−1 this construction. It is convenient to let δ = z so that h Lih = δ hLih. RESTRICTIONS ON POLYNOMIALS ARISING FROM LOCAL MOVES 3 The bracket construction of the Homflypt polynomial is analogous to the standard construction of the Kauffman polynomial FL = FL(a; z), which is defined by a bracket hLik according to the rules: h ik + h ik = z h ik + h ik h ik = a h ik −1 h ik = a h ik h ik = 1 where hLik is also a regular isotopy invariant. The Kauffman polynomial is −w then defined by FL = a hLik, where w is again the writhe of L. We let a+a−1 λ = z − 1 so that h Lik = λ hLik. As usual, the entire knot is not always drawn in a bracket calculation; two diagrams in a calculation should be presumed identical outside of the portion drawn. 3. Local Moves and Homflypt Polynomials In this section we establish the main divisibility criterion for the Homflypt polynomial, namely that the Homflypt polynomials of any two n-component −1 −1 links (n ≥ 1) are equal in a certain quotient of the ring Z[a; a ; z; z ]. 2 −2 2 Recall [9] that for a knot K, we have GK 2 Z[a ; a ; z ], and more gen- erally, the Homflypt polynomial of an n-component link is an element of −1 −1 Z[a; a ; z; z ], with powers of a and z being odd or even when n is even or odd respectively. The lowest power of z is 1 − n. Theorem 1. Let L1 and L2 be n-component links. Then GL1 − GL2 is a multiple of a4 − 2a2 + 1 − a2z2. Proof. We will show that when two diagrams differ by a crossing change, then the difference of their Homflypt polynomials is a multiple of a4 − 2a2 + 1 − a2z2. Since one can convert any link into any other link with the same number of components using a sequence of crossing changes, the result fol- lows. Any oriented link diagram with a crossing can be drawn as one of the diagrams in Figure 2. Label these L1 and L2 respectively, where L1 has the Figure 2. Two links that differ by a crossing change. positive crossing. Let w be the writhe of L1, so w − 2 is the writhe of L2. 4 GANZELL, GONZALEZ, MARCUM, RYALLS, AND SANTOS By repeated use of the definition of GL(a; z), we may write D E D E D E = p0 + p1 ; h h h where p0 and p1 are polynomials in a and z. Thus we have ! −w 2 GL1 − GL2 = a − a h h * + ! = a−w z + (1 − a2) h h −w D E D E = a zp0 + zp1 h h 2 D E 2 D E + (1 − a )p0 + (1 − a )p1 h h −w −1 = a zp0δ + zp1a 2 −1 2 −1 −1 −1 + (1 − a )p0a + (1 − a )p1 (a − a )z − a z −w −1 −1 = a p0(a − a + a − a) −1 −1 −1 −1 3 −1 −1 + p1(za + 2az − a z − a z − a z + az) −w−1 −1 4 2 2 2 = −a z p1 a − 2a + 1 − a z ; as desired. The polynomial a4 − 2a2 + 1 − a2z2 appears throughout this paper. We denote it by Ph(a; z): 4 2 2 2 2 2 (1) Ph(a; z) = a − 2a + 1 − a z = (a − 1 − az)(a − 1 + az): Note that Ph(a; z) is precisely the difference between the Homflypt poly- nomial of the unknot and that of the right-handed trefoil. Thus Ph(a; z) is maximal in the sense that any polynomial that divides the difference of every pair of Homflypt polynomials will divide Ph(a; z). Corollary 2. The Homflypt polynomial of a knot has the general form Ph(a; z)g(a; z) + 1. More generally, the Homflypt polynomial of a link with n−1 n components has the general form Ph(a; z)g(a; z) + δ . Proof. This follows immediately from Theorem 1 since the Homflypt poly- n−1 nomial of the n-component unlink is δ . a2−1 Corollary 3. For any link L, if z = a , then GL(a; z) = 1. Proof. From (1), we see that Ph(a; z) = 0 under the given substitution. a−a−1 Moreover, since δ = z , we have that δ = 1. Thus the general form of Corollary 2 gives us the result. RESTRICTIONS ON POLYNOMIALS ARISING FROM LOCAL MOVES 5 Theorem 1 gives us general information regarding Homflypt polynomials and their derivatives. If we consider Homflypt polynomials to be elements −1 −1 of Z[a; a ] [z; z ], then the leading coefficient (in this z-expansion) is a (Laurent) polynomial in the indeterminate a. Note that for knots these are the terms with z-exponent equal to 0. For a given link L, we will denote the j coefficient of z in the expansion of GL(a; z) by qj(a).